Quantum computing with d-wave superconductors

ABSTRACT

A method and structure for a d-wave qubit structure includes a qubit disk formed at a multi-crystal junction (or qubit ring) and a superconducting screening structure surrounding the qubit. The structure may also include a superconducting sensing loop, where the superconducting sensing loop comprises an s-wave superconducting ring. The structure may also include a superconducting field effect transistor.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention generally relates to superconductors, and moreparticularly to a d-wave quantum bit which permits large-scaleintegration of quantum bits for use with quantum computers.

2. Description of the Related Art

A practical design of a quantum computer requires hundreds to thousandsof quantum bits (“qubits”), but up to now realizations of qubits bymethods such as nuclear magnetic resonance (NMR) seem unsuitable for theminiaturization required to enable a many-qubit machine to beconstructed at reasonable cost (e.g., see Gershenfeld, et al. BulkSpin-Resonance Quantum Computation, Science, Vol 275, pp 350-356 (1997)and Chuang et al. Experimental realization of a quantum algorithm NatureVol 393, pp 143-146 (1998), incorporated herein by reference).

Quantum computers promise enormous speed. However, quantum computing canonly be realized if the quantum computing device (quantum computer,quamputer) is built on a scale of at least several thousand qubits. Theinherent scalability of solid state devices and the high level ofexpertise existing in conventional industrial electronics andexperimental mesoscopic physics make solid state-based quamputers anattractive choice.

Quantum coherence preservation (e.g., maintenance of the quantum statefor any useful time period) within a single qubit, is a major problem,also when several qubits are placed in close proximity, they tend toelectromagnetically interfere with each other and destroy anycharge/signal which is stored in adjacent qubits.

The macroscopic coherent ground state and gapped excitation spectrum insuperconductors are favorable situations for coherence preservation. Asdiscussed in greater detail below, the invention comprises a qubitimplementation in solid state integrated circuit technology which cansupport LSI (Large Scale Integration). The quantum computer chip isoperable at very low (milliKelvin) temperatures, which are required toensure purity of quantum states and to minimize noise.

SUMMARY OF THE INVENTION

It is, therefore, an object of the present invention to provide astructure and method for a d-wave qubit structure that includes a qubitdisk formed at a multi-crystal junction (or a superconducting qubitring) and a superconducting screening structure (e.g., ring or disk)surrounding the qubit. The structure may also include a superconductingsensing loop, where the superconducting sensing loop comprises an s-waveand/or d-wave superconducting ring. The superconducting screening ringmay include at least one weak link, driven to the normal state by ameans such as a superconducting field effect transistor, or a laserbeam.

The multi-crystal junction comprises a junction (e.g. disk) ofdifferently aligned controlled orientation high temperaturesuperconductor crystalline structures or a superconducting ring. Therelative orientations of the grains of the crystalline structures arechosen such that the superconducting screening ring spontaneouslygenerates a half-integer quantum of flux at some or all of the grainboundary intersection points. The superconducting screening ringcomprises one of cuprate, niobium or lead. The invention also includesan array of such quantum bit structures and a quantum computer includingthe quantum bit structures

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, aspects and advantages will be betterunderstood from the following detailed description of a preferredembodiment of the invention with reference to the drawings, in which:

FIG. 1A is a schematic diagram of a perspective view of the inventived-wave qubit;

FIG. 1B is a schematic diagram of a top view of the inventive d-wavequbit formed at one of many intersections of different crystallinestructures;

FIG. 2 is a schematic diagram of a top view of the inventive d-wavequbit;

FIG. 3 is a schematic diagram of a cross-sectional view of the inventived-wave qubit;

FIG. 4 is a schematic illustration of a top view of a possible type ofsubstrate used to form used to form the inventive d-wave qubits;

FIG. 5 is a schematic illustration of a top view of a second possibletype of substrate used to form the inventive d-wave qubits;

FIG. 6 is a graph illustrating classical formation of a plus or minushalf (or approximately half) flux quantum;

FIG. 7 is a graph illustrating the classical twin well potentialprovided by the inventive qubit structure; and

FIG. 8 is a schematic diagram of a quantum computer according to theinvention.

FIG. 9A shows a “d-wave” order parameter; and

FIG. 9B shows a “s-wave” order parameter.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS OF THE INVENTION

Referring now to the drawings, and more particularly to FIG. 1A, aperspective drawing of one embodiment of the inventive d-wavesuperconductor qubit structure 40 is illustrated. FIG. 2 is a top viewof the same structure and FIG. 3 is a cross-sectional view along amiddle portion of line A—A in FIG. 2.

A qubit 9 is located at the junction of differently aligned hightemperature superconductor crystalline structures 1 with grain boundaryJosephson junctions separating the regions 1 of different orientation.This is referred to herein as a “multi-crystal” junction or, forexample, if there are three differently aligned regions it is termedherein a “tricrystal” junction. While this embodiment illustrates threedifferently aligned crystalline structures, as would be known by oneordinarily skilled in the art given this disclosure, any number ofdifferently oriented crystalline structures could be utilized with theinvention.

In a preferred embodiment, the crystalline structure 1 comprises a hightemperature crystalline superconducting material (e.g., (100)SrTiO3(STO) or YSZ (Yttrium stabilized Zirconia)) deposited epitaxiallyon a substrate. As shown in FIGS. 4 and 5, the relative orientations ofthe hexagonal grains 1 are chosen so that when a d-wave cupratesuperconductor 9 (e.g., YBCO Bi2212, etc.) is deposited (epitaxially)upon the multigrain substrate 1, the orientation of the d-wavesuperconducting order parameter, in the differently-oriented crystallineregions, generates a half-integer quantum of flux at some or all of the3-grain boundary interjection points (see Tsuei et al. “Pairing Symmetryand Flux Organization in a Tricrystal Superconducting Ring ofYBa₂Cu₃O⁷⁻⁵”, Phys. Rev. Lett., 73, 593, 1994, and Tsuei et al., “Pured_(x) ²-_(y) ² Order Parameter Symmetry in the Tetragonal SuperconductorTl₂Ba₂CuO₆₊₅”, Nature, 387, 481, 1997, incorporated herein byreference).

The inventive structure also includes a screening isolation ring 2(d-wave or s-wave) which surrounds the crystalline material 1. Thescreening ring 2 comprises a d-wave ring formed of Cuprate, or an s-wavering formed from Niobium (Nb) or Lead (Pb) fabricated, for example, byconventional deposition.

One can achieve a similar array of d-wave qubits by means of abiepitaxial film growth technique. (e.g., K. Char et al., “Bi-epitaxialGrain Boundary Junctions in YBa₂Cu₃O₇”, Appl. Phys. Lett, 59,733 (1991),incorporated herein by reference).

The material is called a “d-wave” superconductor because it has a d-waveorder parameter as shown in FIG. 9A instead of an “s-wave” orderparameter shown in FIG. 9B (which describes the conventional low-Tcsuperconductors such as Pb, Nb, etc.).

The inventive multi-grain structures such as those shown in FIGS. 4 and5 can be used as a platform for the placement of qubits and relatedintegrated circuits for quantum computing. An example (referring to themulti-grain structure of FIG. 5) is shown in FIG. 1B.

The multigrain substrate 1 can be used as a platform for implementingqubits and related integrated circuits (e.g., control circuitry) forquantum computing. A qubit based on half-integer magnetic flux statescan be fabricated by patterning (e.g., removing) the cuprate to form aninsulating region 5 around any of the tri-grain intersection points 9.The screening ring 2 is separated from the crystal structure 1 and bothare formed on the insulator 5 (e.g., oxide substrate) which is devoid ofcuprate. The insulating ring 5 preferably has a mean diameter of 1-20microns, and a 1-20 micron gap between the edge of the crystallinestructurel and the edge of the screening ring 2. The screening ring/loop2 prevents adjacent qubits from interfering with each other. Thecrystalline intersections which will become qubits are defined bymaterial removal (e.g, cuprate etching) which forms the insulatingloop/ring 5.

In addition, a superconductor field effect transistor (SUFET) controlledby a gate electrode 3 is used to activate or deactivate the screeningring 2. The SUFET gate 3 is insulated from the d-wave screening ring 2by a gate insulator 7. Further, a low carrier concentration material ora local constriction in the screening 8 forms the channel of the SUFET.For example, the local carrier material preferably comprises a lowcarrier concentration form of cuprate superconductor. If the gate 3 isnegative, carriers are induced in the channel 8, which makes thescreening ring 2 superconducting. It has been demonstrated (Ahn et al.,“Electrostatic Modulation of Superconductivity in UltrathinGdBa₂Cu₃O_(7−I) Films”, Science, 284, 1152 (1999) incorporated herein byreference, that the channel is then superconducting. When the gate 3 isat zero and positive relative to the channel 8, the channel 8 isinsulating and the screening ring 2 is not superconducting.

A control/sensing loop 4, that is insulated from the screening ring 2 byan insulating layer 6 passes over the d-wave screening ring 2. Thecontrol/sensing loop 4 performs two functions. First, thecontrol/sensing loop 4 senses the flux Φ in the qubit 9. Secondly, thecontrol/sensing loop 4 functions as a single input gate when it appliespulsed magnetic filed to the qubit. In another embodiment, separateloops can be used fro these two functions. It needs to be asuperconductor to successfully sense the small magnetic flux from qubit9, as done in Tsuei et al., supra.

When functioning as a single input gate to the qubit, thecontrol/sensing loop (as shown in FIG. 1A) generates a magnetic fieldperpendicular to the plane of the substrate. To perform certain quantumgate operations, it may be necessary to apply an in-plane magnetic fieldto the qubits. This can be accomplished by using the technique ofmicrostrip and/or stripline, a well-established technology insuperconductory electronic devices.

As discussed above, item 1 is a qubit. This is shown in FIG. 1A as adisk of d-wave superconductor film 1. Alternatively, a smaller ring (ora loop) 15 of d-wave superconductor film positioned within the screeningring 2 can operate as the qubit. The shield ring 2 is in the off-statewhen a laser beam is incident on the weak-link. Alternatively, if aSUFET is used, the shield ring is normally in the off-state, unless anegative voltage is applied to the SUFET 3.

Shown in FIG. 1B are two types of on-chip SQUIDs (superconductingquantum interference device) 11,12 that can be fabricated in themulti-grain substrate. These SQUIDs 11, 12 constitute an important partof the integrated circuits for computing operations. More specifically,the two types of SQUIDs include: π-SQUIDs 11, located at the tricrystalpoint (characterized by a phase-shift of π in the SQUID loop); and0-SQUIDS 12, zero phase-shift which can be located at any point alongany slant grain boundaries in the multi-grain substrates (see FIGS. 4and 5).

As mentioned above, the invention is not limited to the geometricalorientation of the substrate region in FIGS. 4 and 5 and an alternativedesign for an array of qubits is, for example, a composite strontiumtitanate (STO) substrate, which could be formed to include multi-crystalgrains located wherever a qubit is desired.

All of the above discussed structures can be formed using anyconventional fabrication process, such as successive deposition andlithographic patterning, epitaxial processing, etc. Thus, as illustratedabove, the invention comprises a type of qubit (“d-wave qubit”) that issuitable for an integrated circuit environment.

In one form, as illustrated in FIG. 4, the invention comprises an arrayof the inventive d-wave qubits 40 fabricated on an oxide substrate 5. Inthis multi-grain design, a qubit can be formed only at the tricrystalintersecting points indicated by a circle in FIG. 4. Once again, thequbits 9 are formed at the intersection points. In the tricrystalexample discussed above, the crystalline regions are oriented at 120°,30° and 60° to each other, respectively, as shown in FIG. 4.

To the contrary, in FIG. 5, the crystalline regions are oriented at120°, 45°, and 15° to each other, respectively. This creates, at each ofthese tri-crystal intersection points, the inventive d-wave qubit.

In FIG. 4, the misorientation angle between two cuprate grains is 60° or30°. In FIG. 5, the misorientation angle between two cuprate grains is45° or 0° (equivalently 90°). In the multi-grain design shown in FIG. 5,a qubit can be formed at every comer of the hexagons. Such differentorientations provide different reactions between the adjacent qubitswhen the screening ring 2 is removed/deactivated which, as would beknown by one ordinarily skilled in the art given this disclosure, can beused to make different types of logic or memory circuits, etc.

The multiple-Josephson junction d-wave superconductor loops are capableof spontaneously generating a half flux quantum of magnetic field ineither ‘up’ or ‘down’ orientation (see Tsuei et al., supra). In thelanguage of Quantum Mechanics, these states can be written as |up> and|down> Quantum states respectively.

With the inventive qubit design, especially with small overalldimensions, quantum mechanical hybridization between the |up> and |down>magnetic flux states occurs, resulting in a splitting into a groundstate |0>=c(|up>+|down>) and an excited state |1>=c(|up>−|down>) of thering (where c is a normalization constant). This situation is referredto as Quantum Coherence. This two level quantum system can be placed inany coherent combination state (a|up>+b|down>), where a and b areconstants, and constitutes a quantum bit of information or d-wave qubit.The d-wave qubit, unlike the case of a conventional s-wavesuperconducting loop, has the desirable property that, in the absence ofa magnetic field, the eigenstates of the system are exactly 10>=c(|up>+|down>) and 11>=c(|up>−|down>(due to intrinsic time-reversalinvariance).

The key characteristics required of a qubit are (see Bennett, supra, andLoss, Daniel and David P. DiVicenzo. Quantum Computation with QuantumDots, In Physical Review A, Vol. 57, No. Jan. 1, 1998 incorporatedherein by reference), a high degree of quantum phase coherence (seeTable II below) (this is the most essential characteristic), thecapability of being influenced by an external device (e.g., thecontrol/sensing loop 4), the capability of being sensed by an externaldevice (e.g., the control/sensing loop 4), and the capability formultiple qubits to be controllably placed in interaction with eachother.

The invention achieves all of these key requirements. Quantum coherenceis the persistence (e.g., maintenance) of the quantum state over manytimes its intrinsic period. The Josephson junction d-wavesuperconducting loop 40 has adequate quantum coherence (e.g., see TableII, below). Further, the inventive qubit 40 is capable of beinginfluenced by an external magnetic field (see Tsuei et al., 1994, 1997,supra) and of being sensed by a proximate s-wave (or d-wave) Josephsonjunction loop (see Tsuei et al., 1994, 1997, supra).

To implement the inventive d-wave qubit circuit with the existing solidstate microfabrication technology, the invention includes thecontrolled-orientation multigrain substrate 1 (e.g., FIGS. 1 and 2),discussed above. With the design shown in FIGS. 4 and 5, the shortestseparation distance between qubits is determined by the size of thehexagons (of the order of 100 to 1000 microns). The inter-qubitseparation needs to be relatively large of this order (e.g., 500microns) to aid magnetic insolation of the qubits from each other, andto leave room for ancillary components.

In order to maintain their time evolution in an independent manner, thequbits are magnetically isolated from each other by the d-wave or s-wavering 2. However, this magnetic screening can be selectively removed toallow adjacent qubits to communicate, thereby forming a 2-input quantumgate.

More specifically, the superconductivity of two or more adjacentscreening rings 2 may be turned off by the application of a focusedlaser beam, which temporarily converts a selected ring or rings 2 to thenormal state by raising the temperature above the superconducting point,or the application of voltage to the gate 3 of the superconducting fieldeffect transistors of the adjacent qubits (which temporarily disablesthe screening rings 2), or by other means (e.g., see Leggett et al.Dynamics of the Dissipative Two-State System reviews of Modern PhysicsVol 59, No 1 (1987), and references cited therein, all of which areincorporated herein by reference).

When the magnetic isolation is removed, there is a dipole-dipoleinteraction, which (in the described geometry) is proportional to thedot product of the magnetic moments of the two interacting qubits. Thus,without functioning screening rings 2 in place, a conventionalHeisenberg interaction occurs between adjacent qubits, allowinginformation (data) to be transferred between qubits.

FIG. 6 is a graph illustrating formation of a plus or minus half (orapproximately half) flux quantum at the intersection of the line withthe line curve, provided LI_(c)>Φ/2π. The quantum mechanicalhybridization, in the classical twin well potential provided by theinventive qubit structure, is illustrated in FIG. 7. There is a degreeof freedom for which the potential has the twin-well structureillustrated in FIG. 7, and for which the mass is the capacitance of theJosephson junction (e.g., see Table I below and prior discussion). Theinterwell barrier in FIG. 7 depends on the ring inductance. Interwelltunneling (FIG. 7) is important to establishing the splitting betweenthe ground and excited states and this is aided by light mass (lowcapacitance) and low barrier of the superconductor structure.

The invention can be embodied in any number of different types ofsystems (e.g., quantum computers) and executed in any number ofdifferent ways, as would be known by one ordinarily skilled in the art.For example, as illustrated in FIG. 8, a typical configuration ofquantum computer system in accordance with the invention preferably hasa Program Control Computer (PCC) 80 of conventional type such as a PC.One role of the PCC's first function will be to actuate the sensor loops3 (alternatively, separate actuator and sensor loops could be provided)with a precise current and for a precise time to perform the singleinput gate function. For this purpose, on-chip electronics, (e.g.,executed in standard Si circuitry) will perform the required digital toanalog (D to A) gate/laser and filed loop driver function 81.

Secondly, the state of a qubit array 82 will be analyzed by the sensorloops and the on-chip analog to digital (A to D) converters 83. Theinformation will be returned to the PCC 80.

A third PCC function uses the SUFET gates or laser devices to open foran accurately specified time, a pair of selected superconducting guardrings of two selected qubits so as to form a two input quantum gate.

A fourth PCC function, using a measurement protocol including sensor andactuation (when present) loops, calibrates the gate in a way so thatprecise currents and times required to be programmed into the gateoperations are determined. The calibration operation preferably precedesa production run on the quantum computer. Hence, the PCC 80 is useful incontrolling the operation of the quantum computer.

The following treatment of the invention is based on a single Josephsonjunction superconductor ring, which can be performed (without loss ofgenerality) using C to represent total effective junction capacitanceand defining the single junction to have a π-phase shift.

The Larangian for an LCJπ-Ring 2 is as follows. In terms of Flux Φthrough Ring, which is treated as a parallel LCJ circuit, whoseLagrangian is $\begin{matrix}{{L = {{\frac{1}{2}{C\left( \frac{\Phi}{t} \right)}^{2}} - {\frac{\Phi_{0}I_{c}}{2\pi}{\cos \left( \frac{2{\pi\Phi}}{\Phi_{0}} \right)}} - {\frac{1}{2L}\Phi^{2}}}},} & (1)\end{matrix}$

where I_(c) is the critical current of the junction.

The Lagrange equation of motion is $\begin{matrix}{{C\frac{^{2}}{t^{2}}\Phi} = {{I_{c}{\sin \left( \frac{2{\pi\Phi}}{\Phi_{0}} \right)}} - {\Phi/{L.}}}} & (2)\end{matrix}$

A nontrivial stationary solution only exists if${{LI}_{c}\frac{2x}{\Phi_{0}}} > 1$

(see FIG. 6),

This is quantized by $\begin{matrix}{\frac{\Phi}{t} = {\frac{ih}{2\pi \quad C}\frac{\quad}{\Phi}}} & (3)\end{matrix}$

whence the Hamiltonian becomes $\begin{matrix}{H = {{\frac{- h^{2}}{2C}\frac{^{2}}{\Phi^{2}}} + {\frac{\Phi_{0}I_{c}}{2\pi}{\cos \left( \frac{2{\pi\Phi}}{\Phi_{0}} \right)}} + {\frac{1}{2L}{\Phi^{2}.}}}} & (4)\end{matrix}$

Note that C is mass, in both classical and quantum formulations. Withoutthe Josephson term, this is a Harmonic Oscillator with frequency

ω₀=1/{square root over (LC)}  (5)

e.g. if L=20×10⁻¹² H, and C=2.2×10⁻¹⁵ F, then ω₀=4.7×10¹² Hz.

With the Josephson term the potential is as illustrated in FIG. 7.

Consider the rescaled problem in terms of h=H/(ω₀) $\begin{matrix}{h = {{{- \frac{1}{2}}\frac{^{2}}{2{x^{2}}}} + {{\alpha cos}({ax})} + {\frac{1}{2}x^{2}}}} & (6)\end{matrix}$

where $\begin{matrix}{{{\Phi = {bx}};{b^{2} = \frac{h\sqrt{L}}{2\pi \sqrt{C}}}},{{{hence}\quad \alpha^{2}} = {\frac{\left( {2e} \right)^{2}}{h}\sqrt{\frac{L}{C}}}},} & (7)\end{matrix}$

and α=E_(J)/(ω₀), where $E_{J} = {\frac{\Phi_{0}I_{c}}{2\pi}.}$

Taking the variational function

Ψ=exp(−γ(x−x ₀)²)±exp(−γ(x+x ₀)²),  (8)

with upper sign for bonding and lower sign for antibonding state. Forthis trial function, the energy in units of ω₀ is E ω₀=T+V₀+V₁, where$\begin{matrix}{{T = {\frac{\gamma}{2} \mp \frac{2\gamma^{2}x_{0}^{2}s}{1 \pm s}}},{V_{0} = {\frac{1}{8\gamma} + \frac{x_{0}^{2}/2}{1 \pm s}}},{V_{1} = {{{\alpha }^{\frac{- \alpha^{3}}{8\gamma}}\left( \frac{{\cos \left( {\alpha \quad x_{0}} \right)} \pm s}{1 \pm s} \right)}.}}} & (9)\end{matrix}$

The overlap s is given by

s=exp(−2γx ₀ ²)  (10)

This equation is first studied for the approximation where the overlapis neglected. In this approximation the energy is given by$\begin{matrix}{E = {\frac{\gamma}{2} + \frac{1}{8\gamma} + \frac{x_{0}^{2}}{2} + {{\alpha }^{\frac{- \alpha^{2}}{8\gamma}}{{\cos \left( {\alpha \quad x_{0}} \right)}.}}}} & (11)\end{matrix}$

Differentiating w.r.t. x₀ produces $\begin{matrix}{{{x_{0} - {{a\alpha }^{\frac{- \alpha^{2}}{8\gamma}}{\sin \left( {ax}_{0} \right)}}} = 0},} & (12)\end{matrix}$

this is the condition for the classical flux modified by quantumfluctuations. Differentiating w.r.t γ produces $\begin{matrix}{{\frac{1}{2} - \frac{1}{8\gamma^{2}} + {\frac{{a\alpha}^{2}}{8\gamma^{2}}^{\frac{- a^{2}}{8\gamma}}{\cos \left( {ax}_{0} \right)}}} = 0.} & (13)\end{matrix}$

The quantum fluctuation factor $^{\frac{- \alpha^{2}}{8\gamma}}$

which renormalizes α can in all practical cases be replaced by unity.Then with the notation φ₀=αx₀, the overlap factor s which controls theinterwell tunneling rate can be written as $\begin{matrix}{s = ^{{- \sigma^{\frac{1}{2}}}\varphi \sqrt{{\sin \quad \varphi_{0}} - {\varphi_{0}\cos \quad \varphi_{0}}}}} & (14)\end{matrix}$

where σ=(I_(c)/2e)/U_(C) is a ‘Quantum Dot’ parameter, the ratio ofPlanck constant x hopping frequency across the junction to chargingenergy U_(C). When σ is small w.r.t unity, the charging energy U_(C) canbe resolved within the time of a tunneling event (time-energyuncertainty relation) and individual pair tunneling is applicable. Whenthe parameter σ is large, then the collective picture of tunneling bysuperconducting condensate is applicable.

For present purposes, a classic Tsuei-Kirtley situation with collectivetunneling and with the I_(C)L/Φ₀ ratio large w.r.t unity, implying φ₀≈π,will give a very low tunneling rate due to the large negative exponent.Only by a) approaching the pair tunneling limit, say σ=100, from thecollective side, and/or b) by reducing the I_(C)L/Φ₀ ratio so that, sayφ_(0≈π/)3 will the tunneling rate become significant. Physically, thesetwo steps respectively reduce the mass, which is proportional to C, andthe tunneling barrier, which depends on the I_(C)L product.

The following table presents numerical estimates for a couple of caseswith a 10 micron diameter ring. The width of the ring is W and theenergy difference between the bonding and antibonding states is definedas ΔE.

TABLE I W(μ) I_(c)(μA) L(pH) C(fF) {circumflex over (ω)}₀(μeV) ΔE(μeV)1   20 20 4   350 0.5 0.5 20 20 2.2 500 12  

The foregoing Table I illustrates that, with the invention, a reasonablesplitting between the ground and excited states of the qubit isachievable with reasonable system parameters that are capable of beingfabricated by current lithographic processes.

The Leggett approach for treatment of dissipation follows. The presenceof a finite DOS in the gap of a d-wave superconductor, in fact a DOSproportional to energy measured from mid-gap, leads to the possibilityof dissipation associated with tunneling even at very low temperature.The excited state of the twin-well system has the possibility to decayinto excited quasiparticles near the center of the gap.

The transport at junctions between 2d-wave superconductors has beeninvestigated by several authors, in particular by (Leggett et al.,supra), incorporated herein by reference. These authors find anexpression for the normal component of DC current across the junction asfollows $\begin{matrix}{{j_{n} = \frac{ae^{2}V^{3}}{R_{N}\Delta_{0}^{2}}},} & (15)\end{matrix}$

where V is voltage across the junction, R_(N) is the resistance of thejunction, Δ₀ is the maximum gap in k-space, and A is a constant of orderunity. This result might naively be interpreted as V³ in terms of phasespace factors, a factor V arising from the initial DOS going as energy,and another factor V arising from the final DOS. This argument impliesthat there should be an Ohmic current at finite frequency ω,$\begin{matrix}{{{j_{N}(\omega)} = \frac{b{Vh}^{2}\omega^{2}}{{R\Delta D}_{0}^{2}}},} & (16)\end{matrix}$

where the ω² arises from the initial and final state phase space factorsmentioned above, and b is another numerical factor. It is assumed, forthe purposes of the following derivation, that this argument is correct,though a detailed derivation along the lines of BGZ would be desirable.Now the dissipative process needs to be incorporated into the tunnelingformalism, in order to investigate the magnitude of the resultingdecoherence effect (‘T₁ process’).

Since dissipative processes cannot in principle be treated within aLagrangian formalism, the Leggett and Caldeira stratagem is adopted.This approach considers dissipation as arising from the coupling of thequantum system to a bath of a continuous spectrum of quantumoscillators. Consider, to start with, the Lagrangian for an LCR system,this will have the form: $\begin{matrix}{L = {{\frac{C}{2}{\overset{.}{\Phi}}^{2}} + \frac{\Phi^{2}}{2L} + {\sum\limits_{\quad i}{\Phi \quad q_{i}\gamma_{i}}} + {\frac{1}{2}{\sum\limits_{\quad i}{\left( {{\omega_{i}^{2}q_{i}^{2}} + {\overset{.}{q}}_{i}^{2}} \right).}}}}} & (17)\end{matrix}$

where the last term is the Lagrangian for the bath, and the penultimateterm is the coupling between the bath and the LC system. Thecorresponding Lagrange equations of motion are linear, and can be solvedat finite frequency ω to give $\begin{matrix}{{{{- \omega^{2}}{C\Phi}} + \frac{\Phi}{L} + {\sum\limits_{i}\frac{{\Phi\gamma}_{i}^{2}}{\omega^{2} - \omega_{i}^{2}}}} = 0.} & (18)\end{matrix}$

This should compare with the equation for a simple LCR circuit$\begin{matrix}{{{{- \omega^{2}}{C\Phi}} + \frac{\Phi}{L} + \frac{i\omega\Phi}{R}} = 0.} & (19)\end{matrix}$

Comparing the two equations and using the BGZ expression it follows that$\begin{matrix}{{{{Im}{\sum\limits_{i}\frac{\gamma_{i}^{2}}{\omega^{2} - \omega_{i}^{2} + {i\delta}}}} = {b\frac{\hslash^{2}\omega^{3}}{R_{N}\Delta_{0}^{2}}}},} & (20) \\{{\pi {\sum\limits_{i}{\frac{\gamma_{i}^{2}}{2\omega_{i}}{\delta \left( {\omega - \omega_{i}} \right)}}}} = {b{\frac{\hslash^{2}\omega^{3}}{R_{N}\Delta_{0}^{2}}.}}} & (21)\end{matrix}$

This produces an expression for the weighted DOS of the oscillatorsconsistent with BGZ $\begin{matrix}{{\sum\limits_{i}{\gamma_{i}^{2}{\delta \left( {\omega - \omega_{i}} \right)}}} = {b\frac{2}{\pi}{\frac{\hslash^{2}\omega^{4}}{R_{N}\Delta_{0}^{2}}.}}} & (22)\end{matrix}$

Turning now to the quantum system, the tight binding representation isadopted for simplicity, since the bonding/antibonding energy ΔE is verymuch smaller than all other energies, this will also be an accuraterepresentation. Adopting a Fermionic representation, the operator c_(a)is formed for the left hand state of the well, and the operator c_(b)for the right hand state. The Hamiltonian is $\begin{matrix}{{H = {{t\left( {{c_{a}^{+}c_{b}} + {c_{b}^{+}c_{a}}} \right)} + {\sum\limits_{i}{{\lambda_{1_{i}}\left( {b_{i}^{+} + b_{i}} \right)}\left( {n_{a} - n_{b}} \right)}} + {\sum\limits_{i}{\omega_{i}b_{i}^{+}b_{i}}}}},,} & (23)\end{matrix}$

where the b_(i) are the boson operators for the bath. The flux at theclassical minima in the two tight binding states is defined as ±fΦ₀,where f=½ is the ideal half flux quantum state. Equating the couplingterms in the quantum and classical formulations

λ_(i)(b _(i) ⁺ +b _(i))(n _(a) −n _(b))=fΦ₀ q _(i)γ_(i),  (24)

and using the familiar${\left( {b_{i}^{+} + b_{i}} \right) = {\sqrt{\frac{2\omega_{i}}{h^{-}}}q_{i}}},$

produces the following $\begin{matrix}{{\gamma_{i} = \frac{\lambda_{i}\sqrt{2{\omega_{i}/h^{-}}}}{{f\Phi}_{0}}},} & (25)\end{matrix}$

so the expression for the boson DOS becomes $\begin{matrix}{{{\sum\limits_{i}{\lambda_{i}^{2}{\delta \left( {\omega - \omega_{i}} \right)}}} = {{\frac{f^{2}b}{2}\frac{R_{Q}\hslash^{2}\omega^{3}}{R_{N}\Delta_{0}^{2}}} = {\Gamma\omega}^{3}}},} & (26)\end{matrix}$

where R_(Q)=h/e² is the quantum of resistance. The result is a Quantumformulation, and an expression for the weighted DOS of the quantizedbosons, in terms of the parameters of the d-wave Josephson ring.

It turns out that the problem formulated has been extensivelyinvestigated by Leggett et al. This is the ‘superOhmic’ case which isinherently weak coupling. The lifetime broadening can be estimated fromthe Golden Rule for the process where the excited state of the Quantumsystem drops down to the ground state, while emitting a boson of energyΔE: $\begin{matrix}{{\overset{-}{\hslash}/\tau} = {{\sum\limits_{i}{\lambda_{i}^{2}{\delta \left( {{\Delta \quad E} - {\overset{-}{\hslash}\quad \omega_{i}}} \right)}}} = {\frac{f^{2}b}{2}\quad \frac{R_{Q}\Delta \quad E^{3}}{R_{N}\Delta_{0}^{2}}}}} & (27)\end{matrix}$

so the ratio of lifetime broadening to excitation energy of the Quantumsystem is $\begin{matrix}{{\frac{\hslash/\tau}{\Delta \quad E} = {\frac{f^{2}b}{2}\quad \frac{R_{Q}\Delta \quad E^{2}}{R_{N}\Delta_{0}^{2}}}},} & (28)\end{matrix}$

so a rather elegant expression is arrived at, involving the ratio of thejunction conductance to the quantum of conductance, and the ratio of theexcitation energy of the Quantum system to superconducting maximum spacegap squared. Taking the estimates of ΔE found in the above Table, and avalue of 30 meV for the maximum space gap, and b=1, the following isobtained:

TABLE II ΔE(μeV) (/τ)/ΔE  .5 9 × 10⁻¹⁰ 12   5 × 10⁻⁷ 

These estimates are within the frequently quoted value 10⁻⁵ (e.g. seeLoss, Daniel and David P. DiVicenzo, supra), verifying that theinventive Qubit is stable towards T₁processes.

An estimate of the lifetime of the excited state may be obtained fromEq. (28), giving an estimate of the quantum phase coherence. Resultsestimating the quantum coherence are collected in Table II, based on avalue of high temperature superconducting gap, with a maximum gap of 40meV and when coherence times of 10⁻⁹ to 10⁻⁶ cycles can be estimated.This is within the range (10⁻⁵ cycles) of values considered acceptablein Quantum Computer applications (e.g., see Loss, Daniel and David P.DiVicenzo, supra).

Earlier proposals for a Josephson Junction based qubit exist mostimportantly in a recent experimental paper (e.g., see Y. Nakamura etal., Nature 398, 786 (1999)incorporated herein by reference). However,these suggestions involve conventional s-wave superconductors and do notutilize any structure similar to the inventive d-wave qubit.

There is intrinsically a major problem with the use of s-wavesuperconducting Josephson rings, in that the twin-well energy structureof FIG. 7 is replaced by a three well structure with the central wellbeing the deepest. Only by applying a magnetic field with just the rightvalue can twin energy wells be recovered with a conventional s-wavestructure (as shown in FIG. 6). To the contrary, with the inventived-wave qubit, the two wells have exactly the same energy. Therequirement for this tunable magnetic field to be present is a majorcomplication rendering the s-wave based Josephson qubit far morecomplicated and less reliable than the inventive d-wave structure.

While other d-wave superconductor based qubits have been proposed in thepast (e.g., see L. B. loffe et al., “Environmentally decoupled SDS-waveJosephson Junctions for Quantum Computing”, Nature 398, 679 (1999) andA. M. Zagoskin, “A Scalable, tunable Qubit on a clean DND or GrainBoundary -D Junction: cond-mat/9903170, Mar. 10, 1999, both incorporatedherein by reference), these proposals are both quite different from thepresent invention. The concept described in L. B. loffe et al. isfundamentally based on a phenomenon of nonsinusoidal Josephsoncurrent-phase relationship, for which there is practically noexperimental evidence. Moreover, L. B. loffe et al. includes an SNSjunction which has poor Quantum Coherence characteristics. Further, inL. B. loffe et al. no meaningful estimate of the coherence behavior aregiven. Similarly, in A. M. Zagoskin, the qubit states are 0 and 1,instead of being related to the classical ‘up’ and ‘down’ flux states ofan electrode arrangement. Further, these proposals lack independentloops.

As explained above, the invention allows cubic structures to be formedin a high density matrix. Also, the quantum coherence of the inventivecubit is maintained sufficiently to allow high performance operation ofa quantum computer. Additionally, the inventive qubits are capable ofbeing influenced by an external magnetic field so that they canselectively interact with one another.

While the invention has been described in terms of preferredembodiments, those skilled in the art will recognize that the inventioncan be practiced with modification within the spirit and scope of theappended claims.

What is claimed is:
 1. A d-wave qubit structure comprising: a qubitcomprising a multicrystal junction of superconducting crystallinestructures; and a superconducting screening structure surrounding saidqubit, wherein said qubit comprises one of a superconducting ring and asuperconducting multi-crystal junction disk, wherein said multi-crystaljunction disk includes a junction of differently aligned hightemperature superconductor crystalline structures, and wherein relativeorientations of hexagonal grains of said crystalline structures arechosen such that said qubit generates a half-integer quantum of flux ateach grain boundary interjection point.
 2. An array of d-wave qubitstructures, each of said d-wave qubit structures comprising: a qubitcomprising a multicrystal junction of superconducting crystallinestructures; and a superconducting screening structure surrounding saidqubit, wherein said qubit comprises one of a superconducting ring and asuperconducting multi-crystal junction disk, wherein said multi-crystaljunction disk includes a junction of differently aligned hightemperature superconductor crystalline structures, and wherein relativeorientations of hexagonal grains of said crystalline structures arechosen such that said qubit generates a half-integer quantum of flux ateach grain boundary interjection point.
 3. A quantum computer includingat least one array of d-wave qubit structures, each of said d-wave qubitstructures comprising: a qubit comprising a multicrystal junction ofsuperconducting crystalline structures; and a superconducting screeningstructure surrounding said qubit, wherein said qubit comprises one of asuperconducting ring and a superconducting multi-crystal junction disk,wherein said multi-crystal junction disk includes a junction ofdifferently aligned high temperature superconductor crystallinestructures, and wherein relative orientations of hexagonal grains ofsaid crystalline structures are chosen such that said qubit generates ahalf-integer quantum of flux at each grain boundary intersection point.4. A d-wave qubit structure comprising: a qubit comprising a tricrystaljunction of superconducting crystalline structures; and asuperconducting screening structure surrounding said qubit, wherein saidsuperconducting screening structure is removable.
 5. An array of d-wavequbit structures, each of said d-wave qubit structures comprising: aqubit comprising a tricrystal junction of superconducting crystallinestructures; and a superconducting screening structure surrounding saidqubit, wherein said superconducting screening structure is removable. 6.A quantum computer including at least one array of d-wave qubitstructures, each of said d-wave qubit structures comprising: a qubitcomprising a tricrystal junction of superconducting crystallinestructures; and a superconducting screening structure surrounding saidqubit, wherein said superconducting screening structure is removable.